False Chemical cold packs should be stored in a refrigerator.
A chemical cold pack consists of a bag within a bag. The inner bag contains water. The outer bag typically contains urea or ammonium nitrate. When the inner bag is broken, the water mixes with the chemical in the outer bag, dissolving it. As the chemical dissolves, this is an endothermic reaction. As the reaction absorbs energy, the device cools down.
This cooling will occur independent of whether the cold pack is stored in the refrigerator or not.